Integral operators in spaces of summable functions

1. Linear operators in L? spaces.- 1. The space L?.- 1.1. Description of the spaces.- 1.2. Criteria for compactness.- 1.3. Continuous linear functionals and weak convergence.- 1.4. Semi-ordering in the spaces S and L?.- 1.5. Projections and bases of Haar type.- 1.6. Operators in the spaces L?.- 2. Continuous linear operators.- 2.1. Linear operators.- 2.2. Regular operators.- 2.3. The M. Riesz interpolation theorem.- 2.4. Interpolation theorems for regular operators.- 2.5. Classes of L-characteristics of linear operators.- 2.6. On a property of regular operators.- 2.7. The Marcinkiewics interpolation theorem.- 3. Compact linear operators.- 3.1. Compact linear operators.- 3.2. Compactness and adjoint operators.- 3.3. Properties of operators compact in measure.- 3.4. Interpolation properties of compactness.- 3.5. Strongly continuous linear operators.- 2. Continuity and compactness of linear integral operators.- 4. General theorems on continuity on integral operators.- 4.1. Linear integral operators.- 4.2. Regular operators.- 4.3. Example of a non-regular operator.- 4.4. The adjoint operator.- 4.5. Operators with symmetric kernels.- 4.6. Products of integral operators.- 4.7. Truncations of kernels of integral operators.- 5. General theorems on compactness of integral operators.- 5.1. Problem setting.- 5.2. Regular operators acting from Lo to L?0 and from L?0 to L1.- 5.3. Regular operators acting from L?0 to L?0 where 0 < ?0 < 1, 0 < ?0 ? 1.- 5.4. Regular operators acting from L?o to L?o where 0 < < ?0 < 1, ?0 ? 1.- 5.5. Regular integral operators acting from L1 to L?0.- 5.6. Operators with compact majorants.- 5.7. The case of kernels with reinforced singularities.- 5.8. Truncations of kernels of integral operators.- 5.9. Products of integral operators.- 5.10. Compactness of non-regular operators.- 6. Linear Uo-bounded operators.- 6.1. Simplest criteria for continuity of integral operators.- 6.2. Spaces Eu0.- 6.3. General form of u0-bounded operators.- 6.4. Compactness of u0-bounded operators.- 6.5. Compactness of u0-bounded operators acting from L? to L0.- 6.6. Compactness of u0-bounded operators acting from L1 to L? (? > 0).- 6.7. Integral operators acting from L? To c.- 6.8. ?0-Cobounded linear operators.- 6.9. Compactness of ?0-cobounded operators.- 6.10. Interpolation properties of u0-boundedness.- 6.11. On weakly compact operators in l1.- 7. Integral operators with kernels satisfying conditions of kantorovic type.- 7.1. Simplest criteria.- 7.2. Theorems with intermediate conditions.- 7.3. Lemmas.- 7.4. Applications of theorems on adjoint operators.- 7.5. Fundamental theorems.- 7.6. Conditions of ‘Kantorovic’ type.- 7.7. Summability of kernels of integral operators.- 8. Operators of potential type.- 8.1. Definitions.- 8.2. Simplest theorems on continuity and compactness of potentials.- 8.3. Interpolation theorem of Stein-Weiss.- 8.4. Limit theorems on continuity of potentials.- 8.5. Operators of potential type.- 8.6. The logarithmic potential.- 8.7. Iterates of operators of potential type.- 8.8. Generalizations to the case of distinct dimensions.- 8.9. Potentials with respect to non-Lebesgue measure.- 3. Fractional powers of selfadjoint operators.- 9. Splitting of linear operators.- 9.1. Square root of selfadjoint operators.- 9.2. Splitting of an operator.- 9.3. L-Characteristic of a square root.- 9.4. Representation of compact operators.- 9.5. Square root of integral operator.- 9.6. Example.- 9.7. Investigation of integral operators by means of properties of iterated kernels.- 9.8. Remark on Mercers’ theorem.- 10. Fractional powers of bounded operators.- 10.1. The spectral function.- 10.2. Fractional powers of bounded selfadjoint operators.- 10.3. The fundamental theorem.- 10.4. Operators in real spaces.- 10.5. Fractional powers of compact operators.- 10.6. L-Characteristics of fractional powers of operators.- 10.7. Fractional powers of integral operators.- 11. Unbounded selfadjoint operators.- 11.1. Closed operators.- 11.2. Adjoint operators.- 11.3. Integration with respect to spectral functions.- 11.4. The fundamental theorem on spectral representation of unbounded selfadjoint operators.- 11.5. Functions of selfadjoint operators.- 11.6. Commuting selfadjoint operators.- 11.7. Integrals of operator-functions.- 11.8. Integral representation of fractional powers of an operator.- 12. Properties of fractional powers of unbounded operators.- 12.1. Problem setting.- 12.2. The moment inequality for fractional powers.- 12.3. Subordinate operators.- 12.4. Subordination of fractional powers.- 12.5. Heinz’ first inequality.- 12.6. Heinz’ second inequality.- 12.7. Fractional powers of projected operators.- 12.8. On a special class of selfadjoint operators.- 12.9. Theorems on splitting.- 12.10. Theorems on fractional powers.- 12.11. L-Characteristic of fractional powers.- 4. Fractional powers of operators of positive type.- 13. Semi-groups of operators.- 13.1. Vector-functions and operator-functions.- 13.2. Unbounded operators.- 13.3. Resolvents.- 13.4. Definition of a semi-group.- 13.5. Generator of a semi-group.- 13.6. Theorem of Hille-Phillips-Miyadera.- 13.7. Analytic-semi-groups.- 13.8. Estimates for the operators AnT(t).- 14. Fractional powers of positive-type operators.- 14.1. Positive-type operators.- 14.2. Negative fractional powers.- 14.3. Positive fractional powers.- 14.4. A moment inequality.- 14.5. Operators subordinate to fractional powers of a positive- type operator.- 14.6. General theorems on subordination.- 14.7. Estimates for elements of the form BA??x.- 14.8. Comparison of fractional powers of two operators.- 14.9. Fractional powers of positive-type generators.- 14.10. Compactness of fractional powers.- 14.11. Supplementary remarks.- 15. Moment inequalities and L-characteristics of fractional powers.- 15.1. Lorentz spaces.- 15.2. Linear operators.- 15.3. Interpolation theorems.- 15.4. Fundamental theorems.- 15.5. L-Characteristics of fractional powers.- 15.6. One more theorem on compactness.- 16. Fractional powers of elliptic operators.- 16.1. Elliptic differential expressions.- 16.2. Elliptic operators.- 16.3. Positive-type elliptic operators.- 16.4. Multiplicative inequalities and fractional powers of elliptic operators.- 16.5. L-Characteristics of negative fractional powers of elliptic operators.- 16.6. Further theorems.- 16.7. On integral representations of fractional powers of elliptic operators.- 5. Non-linear integral operators.- 17. The superposition operators.- 17.1. On functions which are continuous in one variable.- 17.2. Simplest properties of the superposition operator.- 17.3. Fundamental theorems.- 17.4. Examples.- 17.5. General form of L-characteristics of superposition operators.- 17.6. Uniform continuity of the superposition operator.- 17.7. Improvement of superposition operators.- 17.8. Supplementary remarks.- 18. Conditions for continuity of integral operators.- 18.1. Definitions and simple properties.- 18.2. Conditions for continuity of Uryson operators.- 18.3. General theorem on continuity of Uryson operators.- 18.4. On a property of Uryson operators.- 18.5. Regular Uryson operators.- 18.6. Special examples.- 18.7. Uryson operators with values in the space of bounded functions.- 18.8. On uniform continuity of Uryson operators.- 19. Conditions for complete continuity of an Uryson operator.- 19.1. Problem setting.- 19.2. Hammerstein operators.- 19.3. Complete continuity of regular Uryson operators acting from L0 to L?, ? ? (0, 1].- 19.4. Complete continuity of regular Uryson operators acting from L? to L?, ? > 0, 0 ? ? ? 1.- 19.5. Special criteria for complete continuity.- 19.6. On L-characteristics of Uryson operators.- 19.7. Weakening of singularities.- 19.8. On two criteria for compactness (in measure) of operators.- 19.9. Complete continuity of Uryson operators with values in L0.- 20. Differentiation of non-linear operators.- 20.1. Derivative of a non-linear operator.- 20.2. General form of the derivative of a superposition operator.- 20.3. Conditions for the differentiability of a superposition operator on the whole space.- 20.4. Sufficient criteria for the differentiability of a superposition operator.- 20.5. Differentiability of superposition operators on dense sets.- 20.6. Derivatives of Hammerstein operators.- 20.7. Derivatives of Uryson operators.- 20.8. A general theorem.- 20.9. Partial criteria for differentiability of Uryson operators.- 20.10. Differentiability of Uryson operators at distinguished points.- 20.11. Asymptotic derivatives of non-linear operators.- 20.12. On higher order derivatives.- 6. Some applications.- 21. Equations with completely continuous operators.- 21.1. Linear equations.- 21.2. On approximate solutions of equations.- 21.3. Existence of solutions of non-linear integral equations.- 21.4. Eigenfunctions of non-linear integral operators.- 22. Convergence of Fouriers’ method.- 22.1. General theorems on convergence of Fouriers’ method.- 22.2. Convergence of Fourier series with respect to eigenfunctions of elliptic operators.- 22.3. Fouriers’ method for hyperbolic equations.- 22.4. Fouriers’ method for parabolic equations.- 23. Translation operators along trajectories of differential equations.- 23.1. Linear equations.- 23.2. The Cauchy operator.- 23.3. Non-linear equations.- 23.4. Equations with unbounded non-linearities.- 23.5. The translation operator.- 23.6. Differentiability of the translation operator.- 23.7. The quasi-translation operator.- 23.8. Equations with variable operators.- 23.9. The translation operator and periodic solutions of parabolic equations.- Index of terminologies.- Index of notations.- Author index.